Apparatus, systems, and methods for talbot spectrometers

ABSTRACT

A non-paraxial Talbot spectrometer includes a transmission grating to receive incident light. The grating period of the transmission grating is comparable to the wavelength of interest so as to allow the Talbot spectrometer to operate outside the paraxial limit. Light transmitted through the transmission grating forms periodic Talbot images. A tilted detector is employed to simultaneously sample the Talbot images at various distances along a direction perpendicular to the grating. Spectral information of the incident light can be calculated by taking Fourier transform of the measured Talbot images or by comparing the measured Talbot images with a library of intensity patterns acquired with light sources having known wavelengths.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.16/244,207, filed Jan. 10, 2019, entitled “APPARATUS, SYSTEMS, ANDMETHODS FOR TALBOT SPECTROMETERS,” which is a continuation of U.S.application Ser. No. 15/254,855, filed Sep. 1, 2016, and entitled“APPARATUS, SYSTEMS, AND METHODS FOR TALBOT SPECTROMETERS,” which inturn claims priority, under 35 U.S.C. 119(e), to: U.S. Application No.62/212,636, filed Sep. 1, 2015, entitled “MINIATURE SPECTROMETER BASEDON MID-FIELD DIFFRACTION IMAGING”; U.S. Application No. 62/213,158,filed Sep. 2, 2015, entitled “MINIATURE SPECTROMETER BASED ON MID-FIELDDIFFRACTION IMAGING”; and U.S. Application No. 62/290,120, filed Feb. 2,2016, entitled “MINIATURE, SUB-NANOMETER RESOLUTION TALBOTSPECTROMETER.” Each of these applications is hereby incorporated hereinby reference in its entirety.

BACKGROUND

Conventional free-space optical spectrometers usually rely on thedispersion properties of diffractive elements, such as gratings, toseparate optical frequencies in the far-field. However, in order toachieve high spectral resolution, the spectrometer typically has a verylarge size or a small input aperture that spatially constricts the inputlight. Therefore, there can be a trade-off between resolution, size, and“light-gathering capability” (also referred to as étendue), which isproportional to the effective area of the aperture and the square of thenumerical aperture.

One way to overcome the above constraints uses on-chip spectrometers,which can have lateral dimensions on the order of hundreds of micronsand are very high resolution. But these on-chip spectrometers tend tosuffer from low étendue due to their small input apertures.

Another way uses many filters to spectrally resolve the input signal.For example, one can use narrow-band resonant filters to achieve highresolution, or use broadband filters and employ spectral reconstructiontechniques to resolve features smaller than the bandwidth of thefilters.

A third approach that may address the trade-off between throughput andresolution for conventional diffractive spectrometers is to replace thesmall input aperture with a so-called “coded aperture,” which allows foran increase in throughput. But this approach usually also includessolving an inverse computational problem to construct the spectrum,which can be complex.

SUMMARY

Apparatus, systems, and methods described herein are generally relatedto spectrometers using Talbot effect in non-paraxial regime. In oneexample, a spectrometer includes a grating to diffract incidentradiation so as to form a plurality of Talbot images at intervals alonga direction perpendicular to the grating. The grating has a gratingperiod d about 1 to about 3 times larger than a wavelength λ of theincident radiation. The spectrometer also includes a detector array,disposed at an angle with respect to the grating, to detect at least aportion of the plurality of Talbot images.

In another example, a method of spectrum analysis includes transmittingincident radiation through a grating to generate a plurality of Talbotimages. The grating has a grating period about 1 to about 3 timesgreater than a wavelength of the incident radiation. The method alsoincludes detecting the Talbot images with a detector array tilted withrespect to the grating and estimating the wavelength based at least inpart on the plurality of Talbot images.

In yet another example, a spectrometer includes a grating to diffractincident radiation so as to form a plurality of Talbot images atintervals along a direction perpendicular to the grating. The gratinghas a grating period d about 1 to about 3 times greater than awavelength λ of the incident radiation. The spectrometer also includes adetector array, disposed at an angle with respect to the grating, todetect at least a portion of the plurality of Talbot images. Thedetector array has a proximal end and a distal end. The proximal end isless than 1 mm away from the grating and the distal end is less than 10mm away from the grating. The detector array also has a pixel pitchsubstantially equal to (2m+1)d/2, where m is a positive integer.

It should be appreciated that all combinations of the foregoing conceptsand additional concepts discussed in greater detail below (provided suchconcepts are not mutually inconsistent) are contemplated as being partof the inventive subject matter disclosed herein. In particular, allcombinations of claimed subject matter appearing at the end of thisdisclosure are contemplated as being part of the inventive subjectmatter disclosed herein. It should also be appreciated that terminologyexplicitly employed herein that also may appear in any disclosureincorporated by reference should be accorded a meaning most consistentwith the particular concepts disclosed herein.

BRIEF DESCRIPTION OF THE DRAWINGS

The skilled artisan will understand that the drawings primarily are forillustrative purposes and are not intended to limit the scope of theinventive subject matter described herein. The drawings are notnecessarily to scale; in some instances, various aspects of theinventive subject matter disclosed herein may be shown exaggerated orenlarged in the drawings to facilitate an understanding of differentfeatures. In the drawings, like reference characters generally refer tolike features (e.g., functionally similar and/or structurally similarelements).

FIGS. 1A and 1B show a top view and a side view, respectively, of aspectrometer employing Talbot effect outside the paraxial limit.

FIG. 1C is a photo of the spectrometer shown in FIGS. 1A and 1B.

FIG. 2 illustrates a method of spectral analysis using the Talbot effectin the non-paraxial regime.

FIGS. 3A and 3B show a perspective view and a side view, respectively,of a spectrometer using a one-dimensional binary transmission gratingand a detector array.

FIGS. 4A and 4B show calculated resolution and bandwidth, respectively,of Talbot spectrometers for given pixel lengths and grating periods as afunction of detector angle.

FIGS. 5A and 5B show parameter settings, including pixel length,detector angle, and bandwidth, that can be used to achieve variousresolutions in the Talbot spectrometer shown in FIGS. 3A and 3B.

FIGS. 6A-6D show the calculated performance of Talbot spectrometers withdifferent acceptance angles for collimated incident light.

FIGS. 7A-7D show calculated resolution limits for a Talbot spectrometerwith incident beams over a range of divergence angles.

FIGS. 8A-8D illustrate the data processing of experimental resultsacquired by a Talbot spectrometer.

FIGS. 9A-9L show raw images, bandpass filtered images, simulated images,and corresponding spectra acquired with a Talbot spectrometer atdifferent tilt angles between the detector array and the grating.

FIG. 10 shows calculated and experimental resolutions of a Talbotspectrometer as a function of the tilt angle between the detector andthe grating.

FIG. 11A shows spectra of a tunable laser source swept from 780 to 950nm at a step size of 10 nm measured with a Talbot spectrometer.

FIG. 11B shows a plot illustrating a linear relation between measuredwavelengths and the laser operating wavelengths shown in FIG. 11A.

FIGS. 12A-12C illustrate a simulation model to investigate the influenceof incoherence in the incident light beams on the resolution of Talbotspectrometers.

FIG. 12D shows calculated effective resolution as a function of θ spreadand φ spread in Talbot spectrometers having a camera tilt angle of 30°.

DETAILED DESCRIPTION

Talbot Spectrometers Operating in Non-Paraxial Regime

To address the challenges in conventional spectrometers, apparatus,systems, and methods described herein employ the Talbot effect generatedby a transmission grating to reconstruct spectral information of thelight incident on the transmission grating. An example Talbotspectrometer includes a transmission grating to receive incident light.The grating period of the transmission grating is comparable to thewavelength of interest so as to allow the Talbot spectrometer to operateoutside the paraxial limit. Light transmitted through the transmissiongrating forms a light field including periodic diffraction patterns(also referred to as Talbot images or self-images in this application).The Talbot spectrometer also includes a tilted detector that cansimultaneously measure the light field at various distances. Spectralinformation of the incident light can be calculated by taking Fouriertransform of the measured light field or by comparing the measureddiffraction patterns with a library of intensity patterns acquired withlight sources having known wavelengths.

The detector in the Talbot spectrometer is placed very close to thetransmission grating (e.g., less than 10 mm away from the grating) tomeasure the mid-field diffraction pattern and therefore the entiresystem can be significantly miniaturized. The short distance between thetransmission grating and the detector can also allow the Talbotspectrometer to measure spectra with low temporal coherence. In otherwords, the spectrometer described herein relaxes the requirement oftemporal coherence of incident light. In addition, the pixel pitch ofthe detector can be further configured to be an odd multiple of half ofthe grating period so as to improve effectiveness of measuring theTalbot images.

FIGS. 1A and 1B show a top view and a side view, respectively, of aTalbot spectrometer 100 operating in non-paraxial regime. Thespectrometer 100 includes a grating 110 (e.g., a 1D binary transmissiongrating) to receive input beam 101. The input beam 101 can be, forexample, emitted, scattered, or diffused from a target (also referred toas a light source). Upon diffraction by the grating 110, the incidentlight 101 forms a light field including multiple Talbot images 105 (alsoreferred to as self-images 105). The grating 110 has a grating period dand is positioned such that its grooves are aligned with the y-axis andthe incident light 101 propagates along the z-axis. The Talbot images105 are located in the mid-field diffraction patterns. In addition, theTalbot images 105 are periodic in the x-direction having a period d (thesame as the grating period d), constant in the y-direction, and periodicin the z-direction with period z_(SI). A detector array 120 (e.g., a 2-Dimager) with a pixel pitch p is tilted along the z-axis at an angleθ_(det) with respect to the y-axis, as shown in FIG. 1B. The detectorarray 120 is used to sample the Talbot images 105 at various locationsin the z direction. In operation, the detector 120 can use only aportion 125 of the pixels for measurement. A processor 130 is operablycoupled to the detector array 120 to calculate the spectrum of theincident light 101 based on the Talbot images 105 sampled by thedetector 120.

FIG. 1C is a photo of the spectrometer 100 shown in FIGS. 1A-1B. Toillustrate the compactness of the device, a quarter dollar is placedbeside the spectrometer for comparison. The spectrometer has dimensionsof about 10 mm×10 mm×6 mm and can achieve sub-nanometer resolution. Incontrast, existing Talbot spectrometers usually have resolutions ofabout 20 nm to about 50 nm and the sizes may be as large as tens ofcentimeters, due to the many optical elements in the holographic system.Smaller sizes (e.g., about 40 mm) are also possible when a lens is usedto magnify the self-images onto a detector.

Without being bound by any particular theory, the operation of thespectrometer 100 can be understood using Fresnel diffraction principles.As understood in the art, Fresnel diffraction from the grating 120 cangenerate a periodic pattern of Talbot images 105, which are observablestarting immediately behind the grating 120. Halfway between theself-imaging planes are the phase-inverted imaging planes, at which thelight and dark regions are swapped (see, e.g., the dashed line shown inFIG. 1A). Therefore, the intensity for a point at position (x, y) canalternate from bright to dark as one moves away from the grating 120 inz direction. Without being bounded by any particular theory or mode ofoperation, this repeating pattern arises from the interference from thediffracted beams. For weakly dispersive gratings, which satisfy theparaxial approximation,d>>λ, the self-images appear at Talbot imageplanes spaced apart by the Talbot distance z_(T),

$\begin{matrix}{{z_{T} = {m\frac{2d^{2}}{\lambda}}},} & (1)\end{matrix}$

where m is an integer corresponding to the interfering diffractionorders, λ is the operating wavelength, and d is the period of thediffraction grating. The distance within which the Talbot effect can beobserved is W/tan ϕ, where W is the width of the grating 120 and ϕ isthe angle of the diffracted beam. The region beyond this distance afterthe grating is usually referred to as the far-field regime, where thediffracted beams no longer overlap.

Since the repeating pattern of the Talbot images 105 is inverselyproportional to the wavelength, the Talbot images 105 can be used toperform spectral reconstruction. The Talbot spectrometer 100 can utilizethe Talbot effect by measuring the field intensity as a function ofdistance from the grating 110 at several Talbot image planessimultaneously. The spectrum of the incident light 101 can be thenreconstructed by taking Fourier transform of the periodic pattern.Alternatively, the distance from the grating 110 can also be used toderive the wavelength.

To avoid using moving parts to measure the field, digital holography canbe employed to construct the Talbot pattern or a tilted detector (asused in the spectrometer 100) can be used to simultaneously measure thefield at various distances.

The spectrometer 100 shown in FIGS. 1A-1B uses a tilted detector array120 that allows the sampling of the diffraction pattern 105 at multipleTalbot image planes along the z direction without moving parts. Thespectrum can be obtained by taking the fast Fourier transform (FFT) ofthe detected Talbot pattern 105. The length of each pixel in the zdirection is z_(pix)=p sin θ_(det), and the total distance in z that theimager spans is z_(spec)=N_(Z)z_(pix), where N_(Z) is the number ofpixels in one dimension of the imager. When z_(pix)<<z_(T), the pixelscan be modeled as delta functions sampling the Talbot images 105, andthe spectrometer's resolution and wavelength span are Δk_(T)=2π/z_(spec)and k_(T,max)=π/z_(pix), respectively, where k_(T)=2π/z_(T).

There can be a trade-off between wavelength span and resolution of thespectrometer 100 for a given detector array 120. On the one hand,increasing the tilt angle of the detector array 120 can increase thespectrometer's resolution because the number of self-images sampled isincreased. On the other hand, increasing the tilt angle can alsodecrease the maximum wavelength that can be detected before aliasingoccurs because the sampling period is increased.

Conventional Talbot spectrometers only use the Talbot effect under theparaxial limit, where the grating period d is much larger than theoperating wavelength λ (i.e.,d>>λ). This may allow the acquisition ofsharp replications of the grating self-images. However, operation in theparaxial limit can constrain the minimum size of the resultingspectrometer. In the paraxial limit, the Talbot distance is usually atleast a hundred times larger than the wavelength. Spectralreconstruction using FFT usually uses many periods of the Talbotself-images that are sampled by the imager for high spectral resolution.To resolve wavelengths δλ apart, the minimum distance that the imagersamples is approximately 2d²/δλ, according to Fourier theory. Forexample, a spectrometer with grating period d=10λ operating at λ=500 nmcan be at least 5 cm long to have a spectral resolution of 1 nm.

In contrast, the grating 110 has a period d comparable to the operatingwavelength λ, i.e., d˜λ, in which case there is significantly diffractedpower in only the +1, 0, and −1 diffraction orders. This case isreferred to as non-paraxial regime throughout this application. TheTalbot images 105 may not be sharp replicas of the pattern of thegrating 110. Instead, the Talbot images include smooth sinusoids (see,e.g., FIG. 1A), and appear periodically in z with period:

$\begin{matrix}{{z_{SI} = \frac{\lambda}{1 - \sqrt{1 - \frac{\lambda^{2}}{d^{2}}}}},} & (2)\end{matrix}$

The Talbot images 105 in non-paraxial regime can arise from theinterference of the −1 and/or +1 diffracted beams with the 0 orderdiffracted beam. In the paraxial limit, Equation (2) simplifies to theTalbot distance in Equation (1).

The wavelength can be calculated from the periodicity of the self-images(z_(SI)) from

$\begin{matrix}{\lambda = \frac{2d^{2}z_{SI}}{d^{2} + z_{SI}^{2}}} & (3)\end{matrix}$

When operating in non-paraxial regime, the distance between theself-images is on the order of the wavelength, so a spectrometer ofcomparable resolution can be a hundred times thinner.

To further improve the performance of the non-paraxial, mid-field Talbotspectrometer, the detector array 120 can be configured to resolve theTalbot images 105 without free-space optics such as lenses. In oneexample, the pixel pitch p of the detector array 120 can be differentfrom integer multiple of the grating period d, i.e., p≠nd, where n is apositive integer. Otherwise, the detector array 120 may not be able todetect the difference in intensity at the self-imaging planes and thephase-inverted self-imaging planes, and the recorded pattern may beconstant in z. In another example, the Talbot signal can be increased ormaximized by using a pixel pitch p that is an odd multiple of half thegrating period d, i.e., p=(2m+1)d/2, where m is a positive integer.

Based on the above description, components in the spectrometer 100 canbe configured accordingly. The ratio of the grating 110 period d to thewavelength λ of the incident beam 101 determines the regime of thespectrometer 100 (e.g., paraxial regime or non-paraxial regime). In oneexample, the ratio d/λ can be about 1 to about 3 (e.g., about 1, about1.5, about 2, about 2.5, or about 3, including any values and sub-rangesin between). In another example, the grating period d can be less thanthe wavelength λ to further increase the diffraction effect (e.g., d/λcan be less than 1, less than 0.9, less than 0.8, less than 0.7, lessthan 0.6, less than 0.5, less than 0.4, or less than 0.3, including anyvalues and sub-ranges in between).

In one example, the grating 110 has a constant period d across theentire grating 110. In another example, the grating 110 can have avarying period across the grating 110. For example, the period of thegrating 110 can gradually change from d₁ on one end of the grating to d₂on the other end of the grating. This varying-period grating (alsoreferred to as a varying-pitch grating) may function as a dispersioncompensator, in addition to generating the Talbot images 105.

The absolute value of the grating period d can be application specific.For example, for medical applications, the spectrometer 100 can work inthe visible and near infrared (IR) regions of the electromagneticspectrum, so the grating period d can be about 0.8 μm to about 2 μm. Inanother example, the spectrometer 100, when used in the short wavelengthinfrared (SWIR) regions\ of the electromagnetic spectrum for machinevision, the grating period d can be about 1.2 μm to about 4 μm. Inpractice, the grating period d can be about 0.5 μm to about 5 μm (e.g.,about 0.5 μm, about 0.8 μm, about 1.0 μm, about 1.2 μm, about 1.4 μm,about 1.6 μm, about 1.8 μm, about 2 μm, about 3 μm, about 4 μm, or about5 μm, including any values and sub-ranges in between).

The tilt angle θ_(det) of the detector 120 with respect to the grating110 can influence the resolution of the spectrometer 100. In general, alarger tilt angle θ_(det) can lead to a higher resolution of thespectrometer 100 (see more details below). In practice, the tilt angleθ_(det) can be about 10 degrees to about 40 degrees (e.g., about 10degrees, about 15 degrees, about 20 degrees, about 25 degrees, about 30degrees, about 35 degrees, or about 40 degrees, including any values andsub ranges in between). Tilt angles θ_(det) greater than 40 degrees canalso be used when, for example, the pixel pitch p (or the pixel sizeaccordingly) is small (e.g., less than 1 μm).

The tilt angle θ_(det) can be configured in view of the total length Lof the detector array 120. For example, the tilt angle θ_(det) and thelength L of the detector 120 can be set such that the detector array 120has a projected length along the z direction (i.e., L sin(θ_(det)))greater than the Talbot length z_(SI) (see, Equation (2)) so as tosample at least one Talbot image. In practice, the projected length ofthe detector array 120 can be at least 3 times greater than the Talbotlength (e.g., greater than 3 times, greater than 4 times, greater than 5times, greater than 8 times, greater than 10 times, greater than 12times, greater than 20 times, greater than 50 times, greater than 100times, or greater than 150 times, including any values and sub ranges inbetween). In other words, hundreds of Talbot images can be sampled bythe detector array 120.

As shown in FIG. 1B, the detector array 120 has a proximal end 122 and adistal end 124. Since Talbot images 105 can appear immediately after thegrating 110, the proximal end 122 of the detector array 120 can be veryclose to the grating 110 (e.g., less than 2 mm, less than 1 mm, lessthan 0.8 mm, less than 0.7 mm, or less than 0.6 mm, including any valuesand sub ranges in between). The distal end 124 of the detector 120 canbe about less than 15 mm away from the grating 110 (e.g., less than 15mm, less than 12 mm, less than 10 mm, less than 9 mm, less than 8 mm,less than 7 mm, less than 6 mm, less than 5 mm, or less than 4 mm,including any values and sub ranges in between).

The Talbot images 105 shown in FIG. 1A are periodic along a direction(i.e., z direction) perpendicular to the grating 110 for illustrativepurposes. In practice, the Talbot images 105 can be periodic along anydirection that has a component along the normal to the grating 110.

The processor 130 in the spectrometer 100 is employed to estimate thewavelength(s) of the incident light 101. In one example, the processor130 can perform a Fourier transform of the measured Talbot images 105 toderive the spectral information of the incident light beam 105. TheFourier transform can be a one-dimensional Fast Fourier Transform(1D-FFT) performed on the measured intensity along one column. The 2Dimager also provides a means of reducing noise by averaging the spectrafrom multiple columns of the imager.

In another example, the processor 130 can calculate the spectralinformation of the incident light beam 101 using a trained model bycomparing the detected Talbot images with a library of expectedintensity patterns for a known range of wavelengths. The library ofexpected intensity patterns can be created through a calibrationprocess. In the calibration process, light beams at a range of knownwavelengths (e.g., λ₁, λ₂, . . . , to λ_(N)) can illuminate the grating110 and the corresponding intensity patterns (e.g., I₁, I₁, . . . , toI_(N)) can be recorded by the detector 120. In operation of thespectrometer 100, when the light beam 101 at an unknown wavelength isilluminating the grating 110, the recorded pattern I_(uk) can becompared to the sequence of patterns I₁, I₁, . . . , to I_(N)) todetermine which wavelength (λ₁, λ₂, . . . , to λ_(N)) generates similarpatterns. Alternatively, given this library and the response of thespectrometer 100 from the unknown light source, the spectral compositionof the light source can be determined by solving an inverse matrixproblem. The library components can be at least semi-orthogonal to eachother in order to be resolved with confidence.

Methods of Spectral Analysis Using Talbot Images

FIG. 2 illustrates a method 200 for spectral analysis using Talbotimages. At step 210 in the method 200, incident radiation illuminates anon-paraxial grating that has a grating period d comparable to at leastone wavelength λ of a spectral component of the incident radiation. Forexample, the ratio of the grating period d to the wavelength λ (i.e.,d/λ) can be about 1 to about 3 or any other range described above. Thediffraction of the grating generates multiple Talbot images periodicallylocated after the grating along a z direction.

At step 220 of the method 200, a detector samples the Talbot images atvarious locations along the z direction. In one example, the detectorcan be substantially parallel to the grating and measure one Talbotimage at one location after the grating. A moving stage can be used tomove the detector along the z direction so as to sample the Talbotimages (see, e.g., FIG. 3B). In another example, a detector can samplethe Talbot images at various locations along the z direction by tiltingthe detector with respect to the grating (see, e.g., FIG. 1B). Thisconfiguration can work without a moving stage, thereby further reducingthe complexity of the system.

At step 230 of the method 200, a processor estimates the spectralinformation (such as peak wavelength or spectral density) of theincident radiation based on the Talbot images acquired at step 220. Inone example, the processor can perform a 1D-FFT to derive the spectralinformation. In another example, the processor can compare the acquiredTalbot images with a library of expected intensity patterns to determinethe wavelength(s) of spectral components in the incident radiation.

Analysis of Performance of Talbot Spectrometers in Non-Paraxial Regime

FIGS. 3A and 3B shows a perspective view and a side view, respectively,of a spectroscopic system 300 including a binary 1D transmission grating310 to generate Talbot images 305 in the mid-field diffraction pattern.The grooves of the grating 310 are along y direction and the Talbotimages 304 are periodic along z direction. A detector 320 is placed atan angle with respect to the grating 310 to sample the Talbot images 305at various locations along the z direction. The detector 320 can berotated about the x direction by an actuator 325, thereby allowing theanalysis of spectrometer performance as a function of tilt angle. Inanother example, the actuator 325 can also include a moving stage (alsoreferred to as a translation stage) to move the detector 320 along the zdirection. The system 300 can also be used to investigate the influenceof other parameters, such as grating periodicity, tilt angle, andincidence angle of light beams, on the performance of Talbotspectrometers operating in non-paraxial regime.

Grating Periodicity

The mid-field pattern including the Talbot images 305 behind a 1Dtransmission grating 310 is shown in FIG. 3B. The self-images 305 thatoccur every Talbot distance appear as alternating strips of high and lowintensity lines, corresponding to the teeth of the grating 310. Betweeneach of the self-images are the phase-inverted self-images, which havereversed high and low intensity lines. Therefore, when measuring theTalbot periodicity, one can only look at a single point on the along xand observe how it changes in intensity with propagation distance z.

Without being bound by any particular theory, the field behind thegrating of grating periodicity d can be expressed using Lord Rayleigh'sapproximation of the Talbot effect as:

$\begin{matrix}{{I\left( {x,z} \right)} = {{A_{0}}^{2} + {4{A_{1}}^{2}{\cos \left( {2\; \pi \; {x/d}} \right)}^{2}} + {4A_{0}A_{1}{\cos \left( \frac{2\; \pi \; x}{d} \right)}{\cos \left( {2\; \pi \; {x/z_{T}}} \right)}}}} & (4)\end{matrix}$

where A₀ is the diffraction efficiency of the m=0 diffraction order, andA₁ is the diffraction efficiency of the m=1, −1 diffraction orders.

It can be assumed that the power detected by a square pixel with pixellength p is given by:

$\begin{matrix}{P_{pixel} = {{\int_{y_{0}}^{y_{0} + p}{\int_{x_{0}}^{x_{0} + p}{{I\left( {x,z} \right)}{dxdy}}}} = {p\left\lbrack {{p\; A_{0}^{2}} + {4{A_{1}^{2}\left( {\frac{p}{2} + \frac{{d\; {\sin \left( \frac{4\; {\pi \left( {p + x_{0}} \right)}}{d} \right)}} - {d\; {\sin \left( \frac{4\; \pi \; x_{0}}{d} \right)}}}{8\; \pi}} \right)}} - {\frac{{dA}_{0}A_{1}}{\pi}{\cos \left( {2\; \pi \; {z/z_{T}}} \right)}\left( {{\sin \left( \frac{2\; \pi \; x_{0}}{d} \right)} - {\sin \left( \frac{2\; {\pi \left( {p + x_{0}} \right)}}{d} \right)}} \right)}} \right\rbrack}}} & (5)\end{matrix}$

The term of the equation that provides information about the Talbotdistance is the cos(2πz/z_(T)) term.

At z=z_(T), the relationship between the detected power of the Talboteffect and pixel length p is:

$\begin{matrix}{P_{{pix},{Talbot}} = {\frac{{dA}_{0}A_{1}}{\pi}\left( {{\sin \left( \frac{2\; \pi \; x_{0}}{d} \right)} - {\sin \left( \frac{2\; {\pi \left( {p + x_{0}} \right)}}{d} \right)}} \right.}} & (6)\end{matrix}$

Note that this equation neglects the z dependence that the detector canhave in the x and y direction. This z dependence can be accounted for byusing z=z₀+x tan(θ_(det)) or z=z₀+y tan(θ_(det)), where θ_(det) is theangle the detector 320 is tilted with respect to the x- and/or y-axes.

In one example, when analyzing the contrast along the detector in thex-direction, the detector can be parallel to the transmission grating(θ_(det)=0) and Equation (5) can be sufficient for this purpose.

First, assuming that the self-image is centered on the pixel, such thatx₀=p/2+mp, where m is an arbitrary integer. Then, if the pixel width pis an odd multiple of half of the grating periodicity d, then the pixelsmeasuring the Talbot self-images can see n peaks and n−1 nulls and thepixels measuring the inverted self-images can see n−1 peaks and n nulls.Note that this assumes that the peaks from the self-image are centeredon the pixel. In contrast, if the pixel width is a multiple of thegrating periodicity, each pixel can see n peaks and n nulls. As aresult, intensity detected by each pixel may appear the same, and noTalbot distance may be measured.

In one example, it can be beneficial to set n=1, so that the pixel widthis half the periodicity of the grating and each pixel measures a singlepeak or null. However, if the grating 310 is to only have propagatingzero and first diffraction orders for wavelengths longer than 800 nm,the grating periodicity at maximum can be 1600 nm. The width of eachpeak or null can be approximately 800 nm, and the corresponding pixelpitch of the detector 320 is about just 800 nm, which may be challengingto fabricate.

In another example, one can set n=2, and fit 1.5 periods of theself-image onto a single pixel. This can correspond to a pixel width ofat most 2400 nm, which is a reasonable size in current CMOS detectors.

It can also be beneficial to have good control over x₀ (i.e., the offsetof the imager pixel from the Talbot pattern) by, for example, moving theimage sensor in the x direction so as to increase the amplitude of theTalbot effect. On the other hand, it can be challenging to observe theTalbot effect if p is an integer multiple of d, regardless of x₀. In thecases when p≠m(d/2), where m is an arbitrary integer, the measuredamplitude of the Talbot effect can vary across the detector. This can beadvantageous if precise control of the position of the grating relativeto the detector is inconvenient or unavailable, because it can guaranteethat some signal can be detected.

If the desired relationship between the grating 310 and pixelperiodicities is not available, the grating 310 can be rotated withrespect to the incident beam to reduce the effective grating periodicityto d cos θ, though this may change the diffraction efficiency of thegrating.

Tilt Direction of Detector

In one example, the detector 320 can be tilted parallel to the teeth ofthe grating 310. In another example, the detector 320 can be tiltedperpendicular to the teeth of the grating 310. It can be helpful toinvestigate the difference between these two configurations so as tofurther improve the performance of the system 300.

In FIG. 3A, the teeth of the grating 310 are oriented vertically alongthe y-axis and the detector 320 is rotated about the x-axis to samplethe Talbot distances along each column and the grating periodicity alongeach row. If the detector 320 is instead tilted around the y-axis, thenthe detector samples the Talbot distances and grating periodicitysimultaneously along each row, while no information is collected alongeach column. Therefore, the first option can be more attractive becausethe intensities at the Talbot distances are sampled independently, inwhich case the calculations can be simplified.

In one example, the detector 320 is rotated about the x-axis, then they-axis of the detector 320 can sample the mid-field diffraction patternat z=y tan θ_(det). When p sin θ_(det)<<z_(T), the contrast of themeasured Talbot signal can be comparable to the contrast derived in theprevious section. As the detector tilt increases, the contrastdecreases. When p sin θ_(det)=z_(T), the contrast can be zero.

In another example, the detector 320 is rotated about they-axis, thenthe detector 320 can sample the mid-field diffraction pattern along aline s where s=x cos θ_(det)=z sin θ_(det). Therefore, when θ_(det)=0,the detector can only sample along the x-axis, while when θ_(det)=90°,the detector can sample along the z-axis. The effective pixel length inx is then p_(eff)=p cos θ_(det).

The tilt can allow for the peaks and nulls of the light field tocorrespond to a single pixel along a row of pixels, thereby increasingthe detected Talbot signal. For example, by integrating over the fieldat the detector 320 for each pixel, where z=x tan θ_(det), it can beseen that depending on the tilt of the detector 320, the contrast of thesignal can increase at relatively large tilts. However, unless θ_(det)is close to 0 or 90 degrees, the signal measured at the detector may notbe obviously periodic as a function of the Talbot distance. (Thisanalysis also does not account for possibly decreased detectorefficiency at increased tilt angles.)

Resolution of the Talbot Spectrometer

As described above, the Fourier transform can be used to determine thespectral information, in which case the resolution of the spectrometer300 can depend on the number of periods that the detector 320 samples.The bandwidth of the spectrometer 300 can depend on the samplingfrequency, which can be approximated by the pixel pitch of the detector320.

The detector 320 can be modeled as a finite series of delta functions.The spacing of the delta functions corresponds to the effective pixelpitch a, and the length of the series is determined by the number ofpixels N, or the detector length. This model can be expressed as a trainof delta functions multiplied by a large rectangle function. In theFourier space, the system is a train of delta functions with spacing2π/a that is convolved with a narrow sinc function. (The finite extentof each pixel produces a sinc-like apodization in the Fourier domainthat can be neglected if the pixels' active areas are small enoughrelative to their pitch.) The sampling window can be the entire lengthof the detector 320. The parameter that can determine the spectralresolution is the entire sampling length (or imager length along thedirection normal to the grating), and in the Fourier domain the sincfunction associated with this sampling window. This sampling window canbe very large, compared to the Talbot distance.

In operation, the signal on the detector 320 has a Talbot distancez_(T). This signal is multiplied onto the detector 320 in real space, soit is convolved with the detector 320 in Fourier space. Considering asingle period from −π/a from +π/a in the Fourier space, it is expectedto see two sinc functions at the carrier frequencies k_(T)=±2π/z_(T).The bandwidth of the system can be defined as k_(T)=π/a to avoidaliasing, and the resolution of this effect can be determined by thewidth of the sinc function.

According the Rayleigh criterion, the next resolvable peak is at thezero of the sinc function. The discrete time Fourier transform (DTFT) ofa centered box with N discrete elements is sin(Ω(N/2+½))/sin(Ω/2), whereΩ=ωa. Therefore, the sinc has zeros at ω(N/2+½)=π/a, and is a functionof the number of pixels in each column of the detector 320.

The effective pixel length can be defined as the length of the pixel inz direction, and is equal to p sin(θ), where p is the actual pixellength, and θ is the angle that the detector 320 is tilted by from thevertical axis. Therefore, the resolution is:

$\begin{matrix}{{\Delta \; k_{T}} = \frac{2\; \pi}{\left( {N + 1} \right)p\; \sin \; \theta}} & (7)\end{matrix}$

And the bandwidth is:

$\begin{matrix}{k_{T,\max} = \frac{\pi}{p\; \sin \; \theta}} & (8)\end{matrix}$

Recall that when d>>λ, where d is the period of the grating 310, theapproximation z_(T)=2d²/λ can be made. In this case, the wavelengthresolution is:

$\begin{matrix}{{\Delta \; \lambda} = \frac{2d^{2}}{\left( {N + 1} \right)p\; \sin \; \theta}} & (9)\end{matrix}$

It appears that the minimum resolvable feature decreases with increasingp, so a large pixel size can be desirable. This makes sense becausehaving a longer pixel size can allow the system 300 to sample moreperiods of the Talbot effect. However, the pixel pitch p may beconstrained by grating period d, which is in turn dependent on theoperating wavelength λ. In non-paraxial regime, the grating period d iscomparable to the operating wavelength λ, and the resolution can besolved numerically.

FIGS. 4A-4B show calculated resolution and bandwidth of Talbotspectrometers for given pixel length p and grating period d as afunction of detector angle. In addition, the bandwidth can be solved foranalytically without approximation:

$\begin{matrix}{\lambda_{\max} = {\frac{2d^{2}z_{T,\max}}{d^{2} + z_{T,\max}^{2}} = \frac{d^{2}p\; \sin \; \theta}{d^{2} + {\left( {p\; \sin \; \theta} \right)^{2}/4}}}} & (10)\end{matrix}$

It can be seen that sub-nanometer resolution can be readily achieved inthe system 300. A bandwidth of more than 2 μm is also practical, therebyallowing the use of the system 300 in various applications involvingspectral analysis in visible and infrared regions of the electromagneticspectrum.

Using geometrical arguments, the grating size matching a detector oflength L and width W can also be determined. The Talbot self-images 305occur when the +1, 0, and −1 diffraction orders overlap. But themeasurement of the Talbot distance can be more reliable where either the+1 and 0 diffraction order exist or the −1 and 0 diffraction ordersexist, though the strength of the signal may be weaker.

The minimum grating size can be defined as the size for which the Talboteffect ends just at the farthest point of the detector 320. In thiscase, the grating size is G=2L sin(θ)tan(θ_(diff)), where θ_(diff) isthe far field diffraction angle. The ideal grating size can be definedas the size for which the entire detector 320 is in the region where the+1, 0, and −1 diffraction orders all exist. In this case, the gratingsize is G=2(L sin θ+W/(2 tan θ_(diff)))tan θ_(diff). The minimum heightof the grating is L cos θ, though in practice it can be larger to avoidedge effects.

Using these various constraint equations, the parameters to obtain acertain resolution can be determined. Because the pixel length andgrating period are closely related for optimal performance, all theconstraints can be expressed in terms of pixel length.

FIG. 5A shows curves relating pixel pitch to detector tilt in order toachieve different resolutions. The grating periodicity is assumed to bed=2/3p. FIG. 5B shows the corresponding bandwidth of the system with theparameters for achieving these resolutions. It can be seen that, for agiven relationship between d and G, detectors with smaller pixel sizescan be used to achieve higher resolution.

Generalizing the Talbot Effect

The above analysis uses the Talbot Effect in its ideal form (e.g. planewave incidence, symmetric binary grating). It can also be helpful toinvestigate the robustness of the Talbot effect including practicalconsiderations such as asymmetric gratings and angular sensitivity.

Asymmetric Grating

The diffraction efficiency of each diffraction order m can be given byA_(m). For an asymmetric grating, A₁≠A⁻¹. The field behind the gratingwhere the three diffraction orders interfere is:

E(x,z,t)=A ₀ exp(j(ωt−kz))+A ₁ exp(j(k _(∥,1) x))exp(j(ωt−k _(⊥,1)xz))+A ⁻¹ exp(−j(k _(∥,1) x))exp(j(ωt−k _(⊥,1) xz))   (11)

Assuming the field amplitudes A_(m) are real, the intensity of the fieldis:

I(x,z)=A ₀ ² +A ₁ ² +A ⁻¹ ²+2A ₁ A ⁻¹ cos(2k _(∥,1) x)+2A ₀(A ₁ +A⁻¹)cos(k _(∥,1) x)cos((k−k _(⊥,1))z)+2A ₀(A ₁ −A ⁻¹)sin(k _(∥,1)x)sin((k−k _(⊥,1))z)  (12)

The field intensity is still periodic in z, with a period of z_(T). Fora symmetric grating as discussed above, where A₁=A⁻¹, the self-imagesremain at a constant lateral position. On the other hand, for aperfectly asymmetric grating, where A₁ (or A⁻¹) is zero while the otherterms are non-zero, the self-images shift in x direction as theypropagate in z direction.

To maximize the cosine term, with the constraint that 1=A₀ ²+A₁ ²+A⁻¹ ²,A₀ can be 1/√2, while A₁=A⁻¹=½. The amplitude of the cosine is then √2,while the sine term goes to 0. The DC terms equals 1, so the ratio ofthe Talbot effect to the DC background is √2. To achieve this, a perfectcosine grating can be desirable.

To maximize the sine term, a perfectly asymmetric grating, with A₀=1/√2,and A₁=1/√2, and A⁻¹=0 can be helpful (A₁ ad A⁻¹ are interchangeable).This means that the cosine term has an amplitude of 1 and the sine termhas an amplitude of 1. The DC terms also equals 1. Therefore, themeasured signal from the Talbot effect can be greater with the optimalsymmetric grating. However, given a non-optimized diffraction grating,the diffraction efficiencies can change depending on the angle ofincidence. Talbot signal can be improved by adjusting the orientation ofthe angle with respect to the beam.

Angular Sensitivity

The above analysis for the Talbot Effect assumes that the incident lightis a perfect plane wave. In practice, the light source might not beperfectly collimated, so it can be helpful to investigate thesensitivity of the Talbot effect with respect to the angular spread ofthe beam. Without being bound by any particular theory, intuition fromLord Rayleigh's derivation of the Talbot effect suggests that theself-images point in the direction of the zeroth-order diffraction beam.This can be simulated by applying a phase exp(−jφ) at the grating, whereφ is the angle of incidence. The simulation shows that the self-imagescan be tilted at an angle. In regions with higher diffraction orders,textures within the self-images can be observed.

In k-space (i.e., wave vector space), the valid combinations of thecomponents of k for a propagating beam at a diffractive medium can berepresented by the surface of a sphere, with the origin positioned atthe value of the propagating k-vector. For example, if the beam ispropagating with {circumflex over (k)}=k_(z){circumflex over (z)}, thenthe origin is at x=0, y=0, and z=−k_(z). For simplicity, a 2D system canbe considered. In paraxial approximation, the valid solutions of k arerepresented by a parabola.

The grating puts a constraint on the k_(x) values the propagating beamcan have. As a result, the beam can have a particular k_(z), whichactually corresponds to the Talbot distance. However, for a beam ofangled incidence on the grating, the origin of the circle can beelsewhere, and the same k_(x) constraint may cause the diffracted beamto have two different values for k_(z). If the incident beam has a rangeof angles, the diffracted beam can have a range of k_(z), and so theTalbot images can begin to blur together, thereby rendering itchallenging to extract any information from the mid-field diffractionpattern.

To better understand the behavior of the system with non-collimatedlight sources, the grating can be illuminated (e.g., in a simulation)with a point source some distance away, such that the grating has acertain angle of acceptance. Acceptance angle can be defined as theangle from normal incidence that the point source makes with respect tothe edge of the grating. It can be further assumed that the light sourceis small, so that the response can be represented by a single pointsource. It turns out that for small acceptance angles, having multiplepoint sources does not affect the detected pattern significantly, sincethe phase front at the grating looks approximately the same for eachpoint source.

FIGS. 6A-6D show the calculated performance of Talbot spectrometers withdifferent acceptance angles at about 2-5 degrees. FIG. 6A shows azoomed-in image of the Talbot self-images for a grating with anacceptance angle of 10 degrees. At this level, no difference isimmediately apparent by eye. FIGS. 6B to 6D show FFTs of the simulateddiffraction pattern assuming 0 degree, 4 degree, and 10 degreeacceptance angles.

FIGS. 7A-7D show calculated resolution limits for non-collimatedincident light beams having a divergence angle of about 2-5 degrees inTalbot spectrometers. FIG. 7A shows a zoomed-out image of the Talbotself-images for a grating with an acceptance angle of 10 degrees. Clearwarping of the self-images is apparent. FIG. 7B shows resolution for aN=500 (i.e. the number of periods) grating with various acceptanceangles. The data is taken from the Fourier transform of a single columnof pixels near the center of the grating. FIG. 7C shows the limit ofresolution as a function of grating and detector size. FIG. 7D shows thelimit of resolution as a function of grating size. It shows that at somepoint, the resolution is no longer determined by the size of the gratingbut by the angle of acceptance instead.

From FIGS. 6A-6D and FIGS. 7A-7D, it can be seen that, in general,increasing the angle of acceptance can increase the breadth of thewavelength peak. In particular, the peak tends to be broadened outtowards shorter wavelengths. The FFT images of the entire detector planeshow that the response from a spherical wave is well-defined. Therefore,it may be possible to correct the images to remove the spherical phasefront, and thereby reduce the peak width of the FFT signal.

Experimental Investigation of Talbot Spectrometers in Non-ParaxialRegime

Experimental investigation of Talbot spectrometers in non-paraxialregime uses two Talbot spectrometers. The first spectrometer includes atransmission phase grating with a grating period d=1.6077 microns (e.g.,manufactured by Ibsen Photonics). The detector includes a monochromeCMOS imaging sensor (e.g., Aptina MT9P031) with pixel pitch p=2.2microns. The detector has 2592×1944 pixels and an active imager size of5.70 mm (H)×4.28 mm (V). On top of the detector is a microlens array andis protected by a glass window. The detector also includes a read-outboard from The Imaging Source.

The second spectrometer uses a transmission grating with a gratingperiod d=1.035 microns (e.g., also from Ibsen Photonics). The detectorincludes a monochrome CMOS imaging sensor (e.g., Aptina MT9J003) withpixel pitch p=1.67 microns. The detector has 3872×2764 pixels and anactive imager size of 6.440 mm (H)×4.616 mm (V). As described above, theTalbot signal can be greatest for a grating with a period of 1.11microns, and no Talbot signal may be detected for gratings with a periodof 0.835 or 1.67 microns. This second spectrometer achievessub-nanometer resolution as described below.

In both cases, a tunable Ti:Sapphire laser or a fixed wavelength laseris coupled to a single mode fiber, and the collimated output is passedthrough a 10× beam expander (e.g., a Thorlabs GBE10-B). The finalcollimated beam is more than 3 cm in diameter, and is normally incidenton the grating. To ensure that the imaged area is in the Talbot zone,one edge of the imager is positioned to nearly touch the grating. Thefarthest point of the image sensor can be less than 6 mm from the top ofthe grating surface.

FIGS. 8A-8D illustrate the data processing of experimental resultsacquired using the first spectrometer. FIG. 8A shows the image takenfrom the detector at a wavelength of 818.20 nm. The detector is placedat 45 degree tilt with respect to the grating. The triangle in the upperleft hand corner is the region where all 3 diffraction orders exist. Itis also the region where the Talbot effect exists. FIG. 8B shows theresult of 1D-FFT taken along each column of the detector (or row of theimage as shown in FIG. 8A). The line corresponding to a wavelength ofabout 1300 nm appears in the top half of the image, where the Talbotself-images exist. FIG. 8C shows the mean of the 1D-FFT data across allthe columns of the detector. FIG. 8D shows 2D-FFT of the image revealingan asymmetry in the frequency domain. Peak features in 2D-FFT image maponto 1D-FFT image.

As shown in FIGS. 8A-8D, analysis of the acquired Talbot images caninclude 1D FFT of each detector column, and then taking the mean of allthe columns to reduce the noise in the 1D FFT signal. To convert thex-axis of the FFT data from k_(z) to λ, Equation (13) can be used:

$\begin{matrix}{\lambda = \frac{2{d^{2}\left( {2\; {\pi/k_{z}}} \right)}}{d^{2} + \left( {2\; {\pi/k_{z}}} \right)^{2}}} & (13)\end{matrix}$

When taking the fast Fourier transform along each column of the image,peaks can be identified to correspond to the wavelength of the incidentlight. The resolution of the measured frequency increases with the angleof the detector, since the detector is able to sample more periods ofthe Talbot effect.

FIGS. 9A-9L show raw images, filters images after a bandpass filter,simulated images, and corresponding spectra using the secondspectrometer (d=1.035 microns, p=1.67 microns) with the detector titledat different angles. FIGS. 9A-9D show the raw image (FIG. 9A),bandpass-filtered image (FIG. 9B), simulated image (FIG. 9C), andcorresponding spectrum (FIG. 9D; acquired for a 100 column subsection)with the detector tilted at 6 degrees. FIGS. 9E-9H show the raw image(FIG. 9E), bandpass filtered image (FIG. 9F), simulated image (FIG. 9G),and corresponding spectrum (FIG. 9H) with the detector tilted at 12degrees. FIGS. 9I-9L show the raw image (FIG. 9I), bandpass filteredimage (FIG. 9J), simulated image (FIG. 9K), and corresponding spectrum(FIG. 9L) with the detector tilted at 21 degrees. The operatingwavelength is 830.15 nm. The arrows point to the main peak. In theimages, the x- and z-axes are along the rows and columns, respectively.

FIG. 9A, FIG. 9E, and FIG. 9I show the periodic pattern of theself-images in the raw images. After applying a simple band-pass filterto isolate the dominant frequency component in the 2D Fourier transformof the image, the self-images can be seen clearly in FIG. 9B, FIG. 9F,and FIG. 9J. The band-pass filter can remove the DC background componentand spurious peaks that potentially arise from unwanted artifacts in theimager itself. All images shown are a 50 pixel by 50 pixel subsection ofthe entire image.

To obtain the spectra shown in FIG. 9D, FIG. 9H, and FIG. 9L, a 100column subsection of the full-length image is used to carry out theFourier transform. There are slight shifts in the center frequency ofthe peaks for different columns of the imager, which may be caused bythe wavefront aberration of the input or the non-uniform microlens arrayacross the imager for chief-ray angle correction for imagingapplication. The 1D FFT of the signal is taken along each detectorcolumn in the subsection, and then the mean of the magnitude of the FFTsis taken to reduce noise.

The experimental results in FIGS. 9A-9L are largely consistent withtheoretical model. The spectra shown in FIG. 9D, FIG. 9H, and FIG. 9Leach include two main peaks, which may be explained by the rotation ofthe imaged patterns in the experiments. The diffraction pattern is ofthe form cos(ax) cos(bz). Under rotation of the imager by a small angleϕ, the image of the field now exhibits two spatial frequencies in the zdirection, b(1−ϕ)±aϕ. Therefore, measuring the signal along the columnsof the imager can produce a spectrum with two peaks near the expectedwavelength. Without being bound by any particular theory or mode ofoperation, the rotated diffraction pattern could be caused by thepossibility that the grating and imager are slightly rotated from eachother in the x-y plane, such that the columns of the imager may not beperfectly parallel with the grooves of the grating.

The measured wavelengths in the spectra shown in FIG. 9D, FIG. 9H, andFIG. 9L are also slightly off the operating wavelength at 830.15 nm.This in part is due to the rotation of the observed diffraction patternas discussed above, but may also because of the high sensitivity of thepeak position to the angle of the incidence of the beam. Accuratedetection of the wavelength can be achieved by calibrating thespectrometer, since the offset in predicted wavelength is constant for aset of measurements (see, e.g., FIG. 11).

The peaks themselves in FIG. 9D, FIG. 9H, and FIG. 9L also have smallsplitting for low camera tilt angles. This most likely is because theincident beam is not perfectly normal to the grating. Non-normalincidence in the x-direction can result in slightly differentdiffraction angles for the +1 and −1 beams, which would result in twoslightly different z_(SI) when they interfere with the 0^(th) diffractedbeam. The +1 and −1 beams can also interfere with each other, but theperiodicity in z will be much larger than z_(SI), so it can be ignored.The relationship between the degree of peak splitting and thex-component of the wavevector, k_(x), for small angle of incidence, isapproximately:

$\begin{matrix}{{{1 \pm \frac{k_{x}k_{g}}{k^{2} + k_{g}^{2}}} = \frac{\lambda_{\pm} + {\lambda_{p}\left( {{- 1} + \sqrt{1 - {\lambda_{p}^{2}/d^{2}}}} \right)}}{\lambda_{\pm}\sqrt{1 - {\lambda_{p}^{2}/d^{2}}}}},} & (14)\end{matrix}$

where k_(g)=2π/d is the grating vector, λ_(p) is the center wavelength,and λ_(±) are the wavelengths of the two split peaks. For geometricreasons, θ_(det) affects whether this peak splitting is observed.

For small θ_(det), the imager can mostly sample the region where allthree diffraction orders (−1, 0, 1) exist, so two z_(SI) can be measuredand peak splitting can be observed according to Equation (14). For largeθ_(det), the imager can mostly sample regions where two of the threediffraction orders (±1, 0) exist. When using a subsection of the image,one z_(SI) is measured and no peak splitting is observed. The smallsplitting of the peaks may limit the resolution that can be obtained. InFIG. 9D, the peaks are split by about 3 nm, which corresponds to anincidence angle of 0.03 degrees. However, one could address this issueby using a modified grating, or operating at a higher incidence anglefor which only the 0 and 1 (or −1) diffraction orders exist. In oneexample, the grating can be modified to have very small diffraction for−1 order compared to +1 order (or to have very small diffraction for +1order compared to −1 order). In another example, a large enough gratingcan be used to reduce the concern that all of the diffracted ordersoverlap simultaneously.

FIG. 10 shows calculated and experimental resolutions of the Talbotspectrometers as a function of detector angle. Operating wavelength of830.15 nm. The inset in FIG. 10 shows spectrum of two combined lasersources obtained using the second spectrometer system at θ_(det)=20degrees. The fixed-wavelength laser operates at 829.95 nm and thetunable is at 829.05 nm. The spectrometer can clearly resolvewavelengths that are 0.9 nm apart.

The resolution can be determined by finding the full-width half max(FWHM) of the peaks. As shown in FIG. 10, the spectrometer having thelarger grating period shows an experimental resolution close to thetheoretical resolution. The spectrometer with the smaller grating periodshows the correct trend, but has a lower resolution than expected at lowθ_(det) because of peak splitting.

FIG. 10 also shows that the spectrometers can simultaneously resolvelight from two (mutually incoherent) lasers (see the inset in FIG. 10).A 50/50 fiber optic coupler is used to mix light from a fixed-wavelengthdiode laser operating at 829.95 nm with light from the tunable laser.The combined light is collimated and passed through the beam-expander.

FIG. 11A shows measured spectra as the laser is swept from 780 to 950 nmat a step size of 10 nm, obtained using spectrometer with grating periodd of 1.035 microns, a pixel pitch of 1.67 microns, and agrating-to-sensor angle (θ_(det)) at 20 degrees. Data is normalized tothe bluer peak when the spectra show two peaks. FIG. 11B shows a plotillustrating a linear relation between the measured wavelength andactual laser wavelength.

In theory, the second spectrometer (d=1.035 microns, p=1.67 microns,θ_(det)=20 degrees) can measure wavelengths from about 520 nm up to 1.03microns. The lower bound can be the wavelength at which the seconddiffraction order begins to exist, and the upper bound can be thewavelength before which aliasing occurs. Experimental data shows thatthe second spectrometer can achieve an operating span of at least 170 nm(limited by the range of the Ti:Sapphire laser at shorter wavelengthsand the sensitivity of the imager at longer wavelengths) and aresolution of less than 1 nm.

FIG. 11B shows that the relationship between the measured wavelength (λ)and laser wavelength (λ₀) is about λ[nm]=0.96λ₀ [nm]+34 [nm]. Thebackground noise for the 950 nm line is noticeably high because ofreduced sensitivity of the imager at longer wavelengths. The backgroundnoise for the 820 nm line is high because more signal power is in thespurious peak.

The experimental data described above shows that the non-paraxial Talboteffect can be used as the dispersion mechanism for building a compactspectrometer with high resolution. It can also be beneficial toinvestigate the influence of spatial incoherence on the resolution ofsuch a spectrometer. Theoretical characterization can be performed usinga light beam having an incidence angle spread over the polar andazimuthal directions on the grating for the Talbot spectrometer.

FIGS. 12A-12C show the simulation model. FIG. 12A shows wave vectorscattering due to grating under normal incidence, and FIG. 12B showswave vector scattering due to grating under oblique incidence. FIG. 12Cillustrates the incidence angle spread on the grating surface.

As seen in FIGS. 12A-12C, the Talbot wave vector k_(T) arises due to theinterference between the +1^(st) and −1^(st) order diffracted beams withthe 0th order diffracted beam. An oblique incidence beam can result in ashift in the Talbot wave vector k_(T) compared to the normal incidencecase, and the shift depends on both θ (in the x-z plane) and φ (in they-z plane). If the incidence beam has an angular spread, the ensemble ofthe shift effectively may blur k_(T) and reduce the resolution of thespectrometer. Due to the asymmetry of the wave vector deflectionintroduced by the grating, the effective resolution of the spectrometersuffers differently for θ spread and φ spread.

FIG. 12D shows calculated effective resolution as a function of θ spreadand φ spread for a camera tilt angle of 30°. The resolution under normalincidence is below 0.5 nm at λ=830 nm. The plot suggests that in orderto maintain the resolution around one nanometer in this ideal model, theangular spread θ can be restrained to be within around one degree and φto be within around one hundredths of a degree. This asymmetry of thedependence of effective resolution over θ spread and φ spread for theTalbot spectrometer is similar to that of a conventional spectrometer,where the spectrometer resolution is more sensitive over the widthdirection of the slit than the length direction of the slit due to theasymmetric dispersion from 1D gratings.

In addition to high resolution, Talbot spectrometers described hereincan also have a large étendue to measure most spectroscopic signals.Conventional spectrometers can have an étendue on the order of 10⁻⁴ to10⁻³ mm². The two Talbot spectrometers described above can have anétendue of about 1.3×10⁻⁴ mm², which is estimated using the secondspectrometer with a 21 degree tilt angle and an acceptance angletolerance of 0.007 degrees in the x-direction and 0.5 degrees in they-direction for 1 nm resolution. This can be done by calculating theeffect of non-normal incidence on z_(SI). Therefore, it is expected thatthe Talbot spectrometers described herein can be used for most nearinfrared (NIR) sensing applications, given that the signal is firstcollimated.

CONCLUSION

While various inventive embodiments have been described and illustratedherein, those of ordinary skill in the art will readily envision avariety of other means and/or structures for performing the functionand/or obtaining the results and/or one or more of the advantagesdescribed herein, and each of such variations and/or modifications isdeemed to be within the scope of the inventive embodiments describedherein. More generally, those skilled in the art will readily appreciatethat all parameters, dimensions, materials, and configurations describedherein are meant to be exemplary and that the actual parameters,dimensions, materials, and/or configurations will depend upon thespecific application or applications for which the inventive teachingsis/are used. Those skilled in the art will recognize, or be able toascertain using no more than routine experimentation, many equivalentsto the specific inventive embodiments described herein. It is,therefore, to be understood that the foregoing embodiments are presentedby way of example only and that, within the scope of the appended claimsand equivalents thereto, inventive embodiments may be practicedotherwise than as specifically described and claimed. Inventiveembodiments of the present disclosure are directed to each individualfeature, system, article, material, kit, and/or method described herein.In addition, any combination of two or more such features, systems,articles, materials, kits, and/or methods, if such features, systems,articles, materials, kits, and/or methods are not mutually inconsistent,is included within the inventive scope of the present disclosure.

Also, various inventive concepts may be embodied as one or more methods,of which an example has been provided. The acts performed as part of themethod may be ordered in any suitable way. Accordingly, embodiments maybe constructed in which acts are performed in an order different thanillustrated, which may include performing some acts simultaneously, eventhough shown as sequential acts in illustrative embodiments.

All definitions, as defined and used herein, should be understood tocontrol over dictionary definitions, definitions in documentsincorporated by reference, and/or ordinary meanings of the definedterms.

The indefinite articles “a” and “an,” as used herein in thespecification and in the claims, unless clearly indicated to thecontrary, should be understood to mean “at least one.”

The phrase “and/or,” as used herein in the specification and in theclaims, should be understood to mean “either or both” of the elements soconjoined, i.e., elements that are conjunctively present in some casesand disjunctively present in other cases. Multiple elements listed with“and/or” should be construed in the same fashion, i.e., “one or more” ofthe elements so conjoined. Other elements may optionally be presentother than the elements specifically identified by the “and/or” clause,whether related or unrelated to those elements specifically identified.Thus, as a non-limiting example, a reference to “A and/or B”, when usedin conjunction with open-ended language such as “comprising” can refer,in one embodiment, to A only (optionally including elements other thanB); in another embodiment, to B only (optionally including elementsother than A); in yet another embodiment, to both A and B (optionallyincluding other elements); etc.

As used herein in the specification and in the claims, “or” should beunderstood to have the same meaning as “and/or” as defined above. Forexample, when separating items in a list, “or” or “and/or” shall beinterpreted as being inclusive, i.e., the inclusion of at least one, butalso including more than one, of a number or list of elements, and,optionally, additional unlisted items. Only terms clearly indicated tothe contrary, such as “only one of” or “exactly one of,” or, when usedin the claims, “consisting of,” will refer to the inclusion of exactlyone element of a number or list of elements. In general, the term “or”as used herein shall only be interpreted as indicating exclusivealternatives (i.e. “one or the other but not both”) when preceded byterms of exclusivity, such as “either,” “one of,” “only one of,” or“exactly one of” “Consisting essentially of,” when used in the claims,shall have its ordinary meaning as used in the field of patent law.

As used herein in the specification and in the claims, the phrase “atleast one,” in reference to a list of one or more elements, should beunderstood to mean at least one element selected from any one or more ofthe elements in the list of elements, but not necessarily including atleast one of each and every element specifically listed within the listof elements and not excluding any combinations of elements in the listof elements. This definition also allows that elements may optionally bepresent other than the elements specifically identified within the listof elements to which the phrase “at least one” refers, whether relatedor unrelated to those elements specifically identified. Thus, as anon-limiting example, “at least one of A and B” (or, equivalently, “atleast one of A or B,” or, equivalently “at least one of A and/or B”) canrefer, in one embodiment, to at least one, optionally including morethan one, A, with no B present (and optionally including elements otherthan B); in another embodiment, to at least one, optionally includingmore than one, B, with no A present (and optionally including elementsother than A); in yet another embodiment, to at least one, optionallyincluding more than one, A, and at least one, optionally including morethan one, B (and optionally including other elements); etc.

In the claims, as well as in the specification above, all transitionalphrases such as “comprising,” “including,” “carrying,” “having,”“containing,” “involving,” “holding,” “composed of,” and the like are tobe understood to be open-ended, i.e., to mean including but not limitedto. Only the transitional phrases “consisting of” and “consistingessentially of” shall be closed or semi-closed transitional phrases,respectively, as set forth in the United States Patent Office Manual ofPatent Examining Procedures, Section 2111.03.

1. A spectrometer comprising: a grating to diffract incident radiationso as to form a plurality of Talbot images via the non-paraxial Talboteffect at intervals along a direction perpendicular to the grating, thegrating having a grating period d about 1 to about 3 times larger than awavelength λ of the incident radiation; and a detector array, disposedat an angle with respect to the grating, to detect at least a portion ofthe plurality of Talbot images.
 2. The spectrometer of claim 1, whereinthe grating period d is about 0.8 μm to about 4 μm.
 3. The spectrometerof claim 1, wherein the detector array has a proximal end and a distalend, the proximal end being less than 1 mm away from the grating and thedistal end being less than 10 mm away from the grating.
 4. Thespectrometer of claim 1, wherein the detector array has a projectedlength at least three times greater than a Talbot length defined by theplurality of Talbot images along the direction perpendicular to thegrating.
 5. The spectrometer of claim 1, wherein the angle is about 10degrees to about 40 degrees.
 6. The spectrometer of claim 1, furthercomprising: a processor, operably coupled to the detector, to estimatethe wavelength based at least in part on the at least a portion of theplurality of Talbot images.
 7. The spectrometer of claim 6, wherein theprocessor is configured to estimate the wavelength by taking a Fouriertransform of the at least a portion of the plurality of Talbot images.8. The spectrometer of claim 6, wherein the processor is configured toestimate the wavelength by comparing the plurality of Talbot images witha library of expected intensity patterns for a range of wavelengths. 9.The spectrometer of claim 1, wherein each Talbot image of the pluralityof Talbot images is a smooth sinusoidal representation of the grating.10. A method of spectrum analysis, the method comprising: transmittingincident radiation through a grating to generate a plurality of Talbotimages via the non-paraxial Talbot effect, the grating having a gratingperiod about 1 to about 3 times greater than a wavelength of theincident radiation; detecting the Talbot images with a detector arraytilted with respect to the grating; and estimating the wavelength basedat least in part on the plurality of Talbot images.
 11. The method ofclaim 10, wherein the grating period is about 0.8 μm to about 2 μm. 12.The method of claim 10, wherein detecting the plurality of Talbot imagescomprises: detecting a portion of a first Talbot image in the pluralityof Talbot images at a first distance less than 1 mm from the gratingwith a proximal end of the detector array; and detecting a portion of asecond Talbot image in the plurality of Talbot images at a seconddistance less than 10 mm from the grating with a distal end of thedetector array.
 13. The method of claim 10, wherein detecting theplurality of Talbot images comprises detecting the Talbot images usingthe detector array having a projected length at least three timesgreater than a Talbot length defined by the plurality of Talbot imagesalong a direction perpendicular to the grating.
 14. The method of claim10, wherein detecting the plurality of Talbot images comprises detectingthe Talbot images using the detector array tilted at an angle of about10 degrees to about 40 degrees with respect to the grating.
 15. Themethod of claim 10, wherein estimating the wavelength comprises taking aFourier transform of the plurality of Talbot images.
 16. The method ofclaim 10, wherein estimating the wavelength comprises comparing theplurality of Talbot images with a library of expected intensity patternsfor a range of wavelengths.
 17. The method of claim 10, wherein eachTalbot image of the plurality of Talbot images is a smooth sinusoidalrepresentation of the grating.
 18. A spectrometer comprising: a gratingto diffract incident radiation so as to form a plurality of Talbotimages via the non-paraxial Talbot effect at intervals along a directionperpendicular to the grating, the grating having a grating period dabout 1 to about 3 times larger than a wavelength λ of the incidentradiation, wherein the Talbot images are formed by an overlap of only azero diffraction order and a first diffraction order of the diffractedincident beam; and a detector array, disposed at an angle with respectto the grating, to detect at least a portion of the plurality of Talbotimages.